A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction

Yves Capdeboscq; Michael S. Vogelius

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 1, page 159-173
  • ISSN: 0764-583X

Abstract

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We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.

How to cite

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Capdeboscq, Yves, and Vogelius, Michael S.. "A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.1 (2003): 159-173. <http://eudml.org/doc/245866>.

@article{Capdeboscq2003,
abstract = {We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.},
author = {Capdeboscq, Yves, Vogelius, Michael S.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {voltage perturbations; conductivity inhomogeneities; low volume fraction},
language = {eng},
number = {1},
pages = {159-173},
publisher = {EDP-Sciences},
title = {A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction},
url = {http://eudml.org/doc/245866},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Capdeboscq, Yves
AU - Vogelius, Michael S.
TI - A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 1
SP - 159
EP - 173
AB - We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.
LA - eng
KW - voltage perturbations; conductivity inhomogeneities; low volume fraction
UR - http://eudml.org/doc/245866
ER -

References

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  1. [1] G. Alessandrini, E. Rosset and J.K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement. Proc. Amer. Math. Soc. 128 (2000) 53–64. Zbl0944.35108
  2. [2] H. Ammari and H. Kang, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. Preprint (2002). Zbl1036.35050MR2001663
  3. [3] H. Ammari and J.K. Seo, A new formula for the reconstruction of conductivity inhomogeneities. Preprint (2002). Zbl1040.78008
  4. [4] H. Ammari, S. Moskow and M.S. Vogelius, Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM Control Optim. Calc. Var. 9 (2003) 49–66. Zbl1075.78010
  5. [5] E. Beretta, A. Mukherjee and M.S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area. Z. Angew. Math. Phys. 52 (2001) 543–572. Zbl0974.78006
  6. [6] E. Beretta, E. Francini and M.S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis. Preprint (2002). Zbl1089.78003MR2020923
  7. [7] M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. (to appear). Zbl1016.65079MR1961882
  8. [8] Y. Capdeboscq and M.S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. ESAIM: M2AN (to appear). Zbl1137.35347MR1991198
  9. [9] D.J. Cedio-Fengya, S. Moskow and M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 14 (1998) 553–595. Zbl0916.35132
  10. [10] A. Friedman and M.S. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Ration. Mech. Anal. 105 (1989) 299–326. Zbl0684.35087
  11. [11] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin, Heidelberg, New York (1983). Zbl0562.35001MR737190
  12. [12] H. Kang, J.K. Seo and D. Sheen, The inverse conductivity problem with one measurement: stability and estimation of size. SIAM J. Math. Anal. 28 (1997) 1389–1405. Zbl0888.35131
  13. [13] O. Kwon, J.K. Seo and J-R. Yoon, A real time algorithm for the location search of discontinuous conductivities with one measurement. Comm. Pure Appl. Math. 55 (2002) 1–29. Zbl1032.78005
  14. [14] F. Murat and L. Tartar, H-Convergence, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev and R.V. Kohn Eds., Progress in Nonlinear Differential Equations and Their Applications, Vol. 31, pp. 21–43. Birkhäuser, Boston, Basel, Berlin (1997). Zbl0920.35019
  15. [15] G.C. Papanicolaou, Diffusion in random media, Surveys in Applied Mathematics, Vol. 1, Chap. 3, J.B. Keller, D.W. Mclaughlin and G.C. Papanicolaou Eds., Plenum Press, New York (1995). Zbl0846.60081MR1366209

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